8 Some irrational numbers

Theorem 8.1

\(e\) is irrational

Proof ▶

TODO

Theorem 8.2

\(e ^2\) is irrational

Proof ▶

TODO

Theorem 8.3 Little Lemma

TODO

Proof ▶

TODO

Theorem 8.4

\(e ^4\) is irrational

Proof ▶

TODO

Lemma 8.5

TODO

Proof ▶

TODO

Lemma 8.6

TODO

Proof ▶

TODO

Lemma 8.7

TODO

Proof ▶

TODO

Theorem 8.8

\(e^r\) is irrational for every \(r\in \mathbb {Q}\setminus \{ 0\} \).

Proof ▶

TODO

Theorem 8.9

\(\pi ^2\) is irrational.

Proof ▶

TODO

Theorem 8.10

For every odd integer \(n \ge 3\), the number

\[ A(n) := \frac{1}{\pi }\arccos \left(\frac{1}{\sqrt{n}}\right) \]

is irrational.

Proof ▶

TODO